Finite Group Databasebeta

← #36352687

Subgroups — #36352687

Order 22144.

256 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 55 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012161C211yesmaximal, characteristic
2#26011783C211yesmaximal, characteristic, complemented
3#26011778C211yesmaximal, characteristic, complemented
4#26011235C211yesmaximal, characteristic, complemented
5#26011223C211yesmaximal, characteristic, complemented
6#26011220C211yesmaximal, characteristic, complemented
7#492×C173C211yesmaximal, characteristic, complemented
8C2×#14583825—21no—
9#19149675—21no—
10#19149651—22no—
11#19149650—21no—
12C4×Dic346C2211yescharacteristic
13C4×D692C2211yescharacteristic
14D4×Dic173—21no—
15#19149612—21no—
16#19149610—21no—
17C4⋊#13063436C2211yescharacteristic
18C4⋊#13063436C2211yescharacteristic
19#19149602—21no—
20#19149599—21no—
21C2×SD16×C173—22no—
22C4⋊Q8×C173C2211yescharacteristic
23#127×C173C2211yescharacteristic
24C4×C8×C173C2211yescharacteristic
25#14583825—42no—
26#14583825—42no—
27C2×Dic346D411yescharacteristic
28C2×D692D411yescharacteristic
29C2×C22⋊D173—22no—
30C2×C22⋊D173—22no—
31C22×Dic173—22no—
32C2×C4×D173—21no—
33C2×#13063436—21no—
34C2×#13063436—21no—
35C4⋊Dic173D411yescharacteristic
36C4⋊Dic173D411yescharacteristic
37#14583805—21no—
38C22⋊Dic173—22no—
39#14583803—21no—
40C4×Dic173—21no—
41C2×Q8×C173—21no—
42C2×D4×C173—21no—
43C2×D4×C173—21no—
44C2×D4×C173D412yes—
45SD16×C173—44no—
46C2×C8×C173—21no—
47C4⋊C4×C173—21no—
48C4⋊C4×C173—21no—
49C42×C173C2311yescharacteristic
50C22×D173—21no—
51Dic346—21no—
52Dic346SD1612yescomplemented
53D692—21no—
54D692—21no—
55C22⋊D173—44no—
56C22⋊D173—82no—
57C2×Dic173—21no—
58C2×Dic173—21no—
59C2×Dic173—21no—
60C2×Dic173—44no—
61C4×D173—41no—
62C4×D173—41no—
63#13063436—41no—
64#13063436—41no—
65C23×C173—22no—
66Q8×C173—42no—
67D4×C173—42no—
68D4×C173—41no—
69D4×C173—41no—
70D4×C173—41no—
71D4×C173—41no—
72C2×C4×C173—41no—
73C2×C4×C173C2×D411yescharacteristic
74C2×C4×C173C2×D411yescharacteristic
75C2×C4×C173C2×D411yescharacteristic
76C8×C173—22no—
77C22×C173—42no—
78C22×C173—42no—
79C22×C173—42no—
80C22×C173#12011yescharacteristic
81C2×D173—41no—
82C2×D173—41no—
83C2×D173—41no—
84C4×C173—21no—
85C4×C173—81no—
86C4×C173C2×SD1611yescharacteristic
87C4×C173#13511yescharacteristic
88C4×C173AGL(1,ℤ/8)11yescharacteristic
89C4×C173#13711yescharacteristic
90Dic173—22no—
91Dic173—41no—
92Dic173—41no—
93C2×C173—82no—
94C2×C173#45011yescharacteristic
95C2×C173#45211yescharacteristic
96C2×C173#45311yescharacteristic
97D173—81no—
98C173#163111yescharacteristic, complemented, Sylow
99#1631—1731nomaximal, Sylow
100#550—1731no—
101#492D17311yescharacteristic, complemented
102#477—1731no—
103#476—1731no—
104#331—1731no—
105Q8⋊C8—1731no—
106D4⋊C8—1731no—
107C2×#41—3461no—
108C2×SD16—22no—
109C4⋊Q8C2×D17311yescharacteristic
110#127—3461no—
111#127C2×D17311yescharacteristic
112#121—3462no—
113#121—3461no—
114C4×Q8—1731no—
115C4×D4—3461no—
116C4×D4—1731no—
117#107—3461no—
118#106—3461no—
119C4⋊C8—1731no—
120C4⋊C8—1731no—
121#103—3461no—
122#102—3461no—
123C4×C8C2×D17311yescharacteristic
124#41—6922no—
125#41—6922no—
126C2×Q8—21no—
127C2×Q8—1731no—
128C2×D4—21no—
129C2×D4—21no—
130C2×D4—3462no—
131C2×D4—3462no—
132C2×D4—1731no—
133C2×D4C22⋊D17312yes—
134C22×C4—3462no—
135C22×C4—3461no—
136SD16—44no—
137C2×C8—21no—
138C2×C8—3461no—
139C2×C8—3461no—
140C4⋊C4—21no—
141C4⋊C4—21no—
142C4⋊C4—3461no—
143C4⋊C4—1731no—
144C4⋊C4—1731no—
145C22⋊C4—3462no—
146C22⋊C4—3461no—
147C42—3461no—
148C42C22×D17311yescharacteristic
149C23—22no—
150C23—3461no—
151Q8—42no—
152Q8—3461no—
153Q8—1732no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—41no—
159D4—3461no—
160D4—3461no—
161D4—13842no—
162D4—6924no—
163C2×C4—41no—
164C2×C4—3461no—
165C2×C4—3461no—
166C2×C4—3461no—
167C2×C4—6924no—
168C2×C4—6921no—
169C2×C4—6921no—
170C2×C4D4×D17311yescharacteristic
171C2×C4D4×D17311yescharacteristic
172C2×C4C2×C22⋊D17311yescharacteristic
173C8—22no—
174C8—6921no—
175C8—6921no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—6921no—
180C22—6921no—
181C22—6921no—
182C22#1914964711yescharacteristic
183C4—21no—
184C4—81no—
185C4—3462no—
186C4—6921no—
187C4—6921no—
188C4SD16×D17311yescharacteristic
189C4#1914970011yescharacteristic
190C4#1914970611yescharacteristic
191C4#1914970811yescharacteristic
192C2—82no—
193C2—13841no—
194C2#2601164011yescharacteristic
195C2#2601165011yescharacteristic
196C2#2601165811yescharacteristic
subgroupG/H
centerC22#19149647
commutator, cosocleC42×C173C23
FrattiniC42C22×D173
Fitting#492×C173C2
socleC22×C173#120
prime radical#26012161C2
hypercenter#492D173
hypocenterC173#1631
LCSG ▷ C42×C173 ▷ C22×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. #26011783C2semidirect
3. #26011778C2semidirect
4. #26011235C2semidirect
5. #26011223C2semidirect
6. #26011220C2semidirect
7. #492×C173C2semidirect
52. Dic346SD16semidirect
98. C173#1631semidirect
101. #492D173semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#260121612
#260117832
#260117782
#260112352
#260112232
#260112202
#492×C1732
#1631173